Further results on the signed Italian domination

摘要

A signed Italian dominating function on a graph G = (V, E) is a function f : V -> {-1, 1, 2} satisfying the condition that for every vertex u, f [u] >= 1. The weight of the signed Italian dominating function, f, is the value f ( V) = Sigma(u is an element of V) f (u). The signed Italian dominating number of a graph G, denoted by gamma(s I) (G), is the minimum weight of a signed Italian dominating function on a graph G. In this paper, we prove that for any tree T of order n >= 2, gamma(s I) ( T) >= -n+4/2 and we characterize all trees attaining this bound. In addition, we obtain some results about the signed Italian domination number of some graph operations. Furthermore, we prove that the signed Italian domination problem is NP-Complete for bipartite graphs.